QB_PH_SEC7_005: Band Theory of Solids & Semiconductors

In a 3D non-interacting free electron gas at temperature $T = 0\text{ K}$, how does the Fermi energy $E_F$ scale with the electron number density $n = N/V$?

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Density of States in $k$-space**: In a 3D volume $V$, the volume per $k$-state is $(2\pi)^3/V$. With spin degeneracy factor $g = 2$: $N = 2 \times \frac{\frac{4}{3}\pi k_F^3}{(2\pi)^3 / V} = \frac{V k_F^3}{3\pi^2}$ 2. **Fermi Wavevector**: $n = \frac{N}{V} = \frac{k_F^3}{3\pi^2} \implies k_F = (3\pi^2 n)^{1/3}$ 3. **Fermi Energy**: $E_F = \frac{\hbar^2 k_F^2}{2m} = \frac{\hbar^2}{2m} (3\pi^2 n)^{2/3}$ 4. **Scaling**: $E_F \propto n^{2/3}$. Option (B) is correct.